Graphs are ubiquitous in all modern sciences. This motivates a need for algorithmic tools that are capable of analyzing and computing on graphs in an efficient manner. A need that is even more pressing now that the graphs we are dealing with tend to be massive, rendering traditional methods no longer adequate.

In this talk, I will discuss a recent progress on the maximum flow problem and use it as an illustration of a broader emerging theme in graph algorithms that employs optimization methods as an algorithmic bridge between our combinatorial and spectral understanding of graphs.

I will also briefly outline how this line of work brings a new perspective on some of the core continuous optimization primitives – most notably, interior-point methods.